= Solution
{c}
For a normalized <orbital> $\int d^2r\,|\phi|^2=1$, the symmetric <Hartree product state> and its equivalent <Fock space> form are
$$
\boxed{\Phi(\mathbf r_1,\ldots,\mathbf r_N)=\prod_{j=1}^N\phi(\mathbf r_j),\qquad
|N:\phi\rangle=\frac{(a_\phi^\dagger)^N}{\sqrt{N!}}|0\rangle,\quad
a_\phi^\dagger=\int d^2r\,\phi(\mathbf r)\psi^\dagger(\mathbf r).}
$$
There are exactly $N(N-1)/2$ interacting pairs, so the <expectation value> of the grand <Hamiltonian> is the <functional>
$$
\boxed{\mathcal E_\mu[\phi]=N\int d^2r\left[\frac{\hbar^2}{2m}|\nabla\phi|^2+\frac\kappa2r^2|\phi|^2\right]
+\frac{gN(N-1)}2\int d^2r\,|\phi|^4-\mu N.}
$$
At fixed $N$ the term $-\mu N$ is constant. Enforce <wavefunction normalization> separately by varying $\mathcal E_\mu-N\eta(\int|\phi|^2-1)$ with respect to $\phi^*$. The resulting stationary <Gross–Pitaevskii equation> for the normalized <orbital> is
$$
\boxed{\left[-\frac{\hbar^2}{2m}\nabla^2+\frac\kappa2r^2+g(N-1)|\phi|^2\right]\phi=\eta\phi.}
$$
The <Lagrange multiplier> $\eta$ enforces the norm; fixed-$N$ variation cannot identify it with the independently inserted <chemical potential> merely by differentiating the constant $-\mu N$.
For the standard large-$N$ convention put $\Psi=\sqrt N\phi$, so $n=|\Psi|^2$ and $\int n=N$, and replace $N-1$ by $N$. Varying the resulting grand <energy> <functional>
$$
\int d^2r\left[\frac{\hbar^2}{2m}|\nabla\Psi|^2+\left(\frac\kappa2r^2-\mu\right)|\Psi|^2+\frac g2|\Psi|^4\right]
$$
with unrestricted $\Psi$ gives
$$
\boxed{\left[-\frac{\hbar^2}{2m}\nabla^2+\frac\kappa2r^2+g|\Psi|^2\right]\Psi=\mu\Psi.}
$$
Equivalently, at fixed $N$ this same $\mu$ is chosen as the normalization <Lagrange multiplier>. This is the <chemical potential> convention used below. The <contact interaction> coupling $g$ has dimensions of <energy> times area in two dimensions.
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